Random perfect lattices and the sphere packing problem

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 86, Issue 4, 2012, Pages

  Andreanov, A ^a   Scardicchio, A ^a,b  

^a ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS   (Italy)

^b SEZIONE DI TRIESTE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

CONSTANT SCALING; EFFECTIVE TEMPERATURE; EUCLIDEAN SPACES; GLASSY SYSTEMS; LARGE DIMENSIONS; LATTICE PACKINGS; LOCAL STRUCTURE; MONTE CARLO SIMULATION; PACKING FRACTIONS; PERFECT LATTICE; RANDOM WALK; SCALE FREE NETWORKS; SPHERE PACKINGS; SPHERE-PACKING PROBLEM;

MONTE CARLO METHODS; PACKING;

CHANNEL CODING;

EID: 84867495698     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.86.041117     Document Type: Article

References (51)

1. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Vol. 290 (Springer, New York, 1999).
2. C. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
3. S. Torquato, Random Heterogeneous Materials (Springer, New York, 2005).
4. L. Fejes Tóth, Math. Z. 0025-5874 10.1007/BF01181430 46, 83 (1940).
5. T. Hales, Ann. Math. ANMAAH 0003-486X 10.4007/annals.2005.162.1065 162, 1065 (2005).
6. S. Torquato and F. H. Stillinger, Exp. Math. 1058-6458 10.1080/10586458.2006.10128964 15, 307 (2006).
7. C. Zachary and S. Torquato, J. Stat. Mech. 1742-5468 10.1088/1742-5468/ 2011/10/P10017 (2011) P10017.
8. A. Scardicchio, F. H. Stillinger, and S. Torquato, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.2897027 49, 043301 (2008).
9. C. Rogers, Proc. Lond. Math. Soc. PLMTAL 0024-6115 10.1112/plms/s3-8.4. 609 3, 609 (1958).
10. G. Parisi, J. Stat. Phys. JSTPBS 0022-4715 10.1007/s10955-008-9539-6 132, 207 (2008).
11. Y. Jin, P. Charbonneau, S. Meyer, C. Song, and F. Zamponi, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.82.051126 82, 051126 (2010).
12. G. Parisi and F. Zamponi, J. Stat. Mech. 1742-5468 10.1088/1742-5468/ 2006/03/P03017 (2006) P03017.
13. G. Parisi and F. Zamponi, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.82.789 82, 789 (2010).
14. H. Cohn and N. Elkies, Ann. Math. 0003-486X 10.4007/annals.2003.157.689 157, 689 (2003).
15. H. Cohn and A. Kumar, Ann. Math. 0003-486X 10.4007/annals.2009.170.1003 170, 1003 (2009).
16. G. Voronoi, J. Reine. Angew. Math. 133, 97 (1908).
17. A. Schürmann, Computational Geometry of Positive Definite Quadratic Forms: Polyhedral Reduction Theories, Algorithms, and Applications, Vol. 48 (American Mathematical Society, Providence, RI, 2009).
18. A. Ash, Can. J. Math CJMAAB 0008-414X 10.4153/CJM-1977-101-2 29, 1040 (1977).
19. J. Lagrange, Nouv. Mém. Acad. Berlin 265 (1773).
20. C. Gauss, J. Reine. Angew. Math. 20, 312 (1840).
21. A. Korkin and E. Zolotarev, Math. Ann. MAANA3 0025-5831 10.1007/BF01442667 11, 242 (1877).
22. E. Barnes, Philos. Trans. R. Soc. Lond. A PTRMAD 1364-503X 10.1098/rsta.1957.0005 249, 461 (1957).
23. D. Jaquet-Chiffelle, Ann. Inst. Fourier (Grenoble) AIFUA7 0373-0956 10.5802/aif.1320 43, 21 (1993).
24. M. D. Sikirić, A. Schürmann, and F. Vallentin, Electron. Res. Announc. Am. Math. Soc 1079-6762 10.1090/S1079-6762-07-00171-0 13, 21 (2007).
25. C. Riener, J. Theorie Nombres Bordeaux 1246-7405 10.5802/jtnb.565 18, 677 (2006).
26. A. Schürmann, fma2.math.uni-magdeburg.de/~latgeo/ and http://www.geometrie.uni-rostock.de/.
27. W. Plesken and B. Souvignier, J. Symb. Comput. 0747-7171 10.1006/jsco.1996.0130 24, 327 (1997).
28. M. Mezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond, Vol. 9 (World Scientific, Singapore, 1987).
29. A. Schürmann, Advances in Mathematics ADMTA4 0001-8708 10.1016/j.aim.2010.05.002 225, 2546 (2010).
30. C. Batut, Math. Comput. 70, 395 (2001).
31. A. Bergé and J. Martinet, J. Lond. Math. Soc. JLMSAK 0024-6107 10.1112/jlms/53.3.417 53, 417 (1996).
32. J. Martinet, Perfect Lattices in Euclidean Spaces, Vol. 327 (Springer, New York, 2003).
33. D. Bremner, M. D. Sikirić, and A. Schürmann, CRM Proceedings & Lecture Notes, 45 (American Mathematical Society, Providence, RI, 2009).
34. D. Avis, D. Bremner, and A. Deza, Polyhedral Computation, Vol. 48 (American Mathematical Society, Providence, RI, 2009).
35. A. Lenstra, H. Lenstra, and L. Lovász, Math. Ann. MAANA3 0025-5831 10.1007/BF01457454 261, 515 (1982).
36. B. Bollobás and O. Riordan, Combinatorica COMBDI 0209-9683 10.1007/s00493-004-0002-2 24, 5 (2004). (Pubitemid 38706499)
37. B. D. Lubachevsky and F. H. Stillinger, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF01025983 60, 561 (1990);
38. B.D. Lubachevsky, F. H. Stillinger, and E. N. Pinson, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF01048304 64, 501 (1991).
39. H. Cohn, A. Kumar, and A. Schürmann, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.061116 80, 061116 (2009).
40. Y. Kallus, V. Elser, and S. Gravel, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.82.056707 82, 056707 (2010).
41. S. Torquato and Y. Jiao, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.82.061302 82, 061302 (2010).
42. M. D. Sikirić, A. Schürmann, and F. Vallentin, Math. Comput. MCMPAF 0025-5718 10.1090/S0025-5718-09-02224-8 78, 1713 (2009).
43. M. D. Sikirić, A. Schürmann, and F. Vallentin, Discrete Comput. Geom. 0179-5376 10.1007/s00454-010-9266-z 44, 904 (2010).
44. T. Rehn and A. Schürmann, Mathematical Sofware ICMS 2010, Lecture Notes in Comput. Sci., Vol. 6327 (Springer, Berlin, 2010), pp. 295-298.
45. The PARI Group, Bordeaux, PARI/GP, version 2.5.2 (2012), http://pari.math.u-bordeaux.fr/.
46. U. Fincke and M. Pohst, Math. Comput. 0025-5718 10.1090/S0025-5718-1985- 0777278-8 44, 463 (1985).
47. H. Cohen, A Course in Computational Algebraic Number Theory, Vol. 138 (Springer, New York, 1993).
48. C. A. Rogers, Acta Mathematica ACMAA8 0001-5962 10.1007/BF02392493 94, 249 (1955).
49. C. A. Rogers, Proc. Lond. Math. Soc. PLMTAL 0024-6115 10.1112/plms/s3-6.2.305 3, 305 (1956).
50. C. A. Rogers, Packing and Covering, Cambridge Mathematical Tracts, No. 54 (Cambridge University Press, Cambridge, 1964).
51. C. Siegel, Ann. Math. ANMAAH 0003-486X 10.2307/1969027 46, 340 (1945).